Collège de France abroad Lectures Quantum information with real or artificial atoms and photons in cavities Lecture 4: Quantum feedback experiments in Cavity QED
Aim of lecture: illustrate on a Cavity QED example the quantum feedback procedure: how to combine measurements and actuator actions on a quantum system to drive it towards a predetermined state and protect this state agaisnt decoherence
IV-A Introduction: principle of quantum feedback Environment Sensor P System in d initial state (known) Target state Actuator Actuator A Controller K A system S coupled to an environment E is initially in a state | ! i >. The goal is to drive it to a target state | ! t > . An actuator A coupled to S transforms its state. Then a sensor (P) performs a measurement sent to a controller (K) which estimates the new state, taking measurement and known effect of environment into account. A distance d to target is computed and K determines the action A should perform to minimize this distance d. Operation repeated in loop until target is reached.
Comparison with classical feedback S-S 0 ? A measurement on system is perfomed (A) and result S is compared to a reference value ! S 0 . A feedback signal -k (S-S 0 ) (where -k is the negative ! k ( S ! S 0 ) gain of the loop) is applied to the system (B) to bring it closer to the ideal operating point. The device operates in closed feedback loop. The feedback can be based on an automatic physical effect with a device combining the measurement and the response mechanism: the Watt regulator of the steam machine is a good example. In other cases, the feedback implies two separate ingredients: a reading apparatus which measures the error signal and an actuator of the response, the link between the two being made by a computer (example: speed controller in an automobile). The extension of these ideas to a quantum system must incorporate an essential element: the measurement has a back-action, independent of any added feedback effect, on the system’s state. This quantum back-action must be taken into account to implement the quantum feedback.
Applying quantum feedback to the stabilization of Fock states? Fock states are interesting examples of non-classical states They are fragile and lose their non-classicality in time scaling as 1/n. The preparation by projective measurement is random Is it possible to prepare them in a deterministic way by using quantum feedback procedures? Can these procedures protect them against quantum jumps (loss or gain of photons)? An ideal sensor for these experiments: QND probe atoms measuring photon number by Ramsey interferometry (see lecture 2). This probe leaves the target state invariant! What kind of actuator? Classical or quantum?
Quantum feedback with classical actuator
Quantum feedback with quantum actuator; atoms not only probe the field (dispersively), but also emit or absorb photons (resonantly)
IV-B Quantum feedback by classical field injections
Principle of quantum feedback by field injections in Cavity Quantum electrodynamics Components of feedback loop Sensor (quantum (quantum “ “eye eye” ”): ): ! Sensor ! atoms and QND measurements atoms and QND measurements ! Contr Contro oller ller ( (“ “brain brain” ”): ): ! computer computer ! Actuator Actuator (classical (classical “ “hand hand” ”): ): ! microwave injection microwave injection Feedback protocol: Inject Inject an initial an initial coherent field coherent field in C in C ! ! Send atoms Send atoms one by one in one by one in Ramsey interferometer Ramsey interferometer ! ! ! Detect each atom Detect each atom, , projecting field density operator projecting field density operator " " in new state in new state estimated estimated by computer by computer ! ! Compute displacement Compute displacement # # which which minimises minimises distance distance D D between target and between target and new state new state ! ! Close feedback Close feedback loop loop by by injecting injecting a a coherent field with coherent field with amplitude amplitude # # in C in C ! Repeat loop until reaching Repeat loop until reaching D ~ 0. D ~ 0. ! !
The CQED Ramsey Interferometer The Ramsey interferometer is made of two auxiliary cavities R 1 et R 2 sandwiching the cavity C containing the field to be measured. The atom with two levels g and e (qubit in states j=0 and j=1 respectively), prepared in e, is submitted to classical $ /2 pulses in R 1 and R 2 , the second having a % r phase difference with the first. The probabilities to detect the atom in g (j=0) and e (j=1) when C is empty are: ( ) P j = cos 2 ! r " j # j = 0,1 ; (7 " 1) 2 The P j probabilities oscillate ideally between 0 and 1 with opposite phases when % r is swept (Ramsey fringes). g , P P e ! r 0 2 !
Single atom detection atom detection ( (see see lecture 2) lecture 2) Single Initial state Atomic detection changes Atomic detection changes the the photon number distribution photon number distribution 0,25 State after 0,20 projection Distribution 0,15 atom in | | e atom in e & & 0,10 direction of measurement 0,05 0,00 0 1 2 3 4 5 6 7 8 Photon number, 0,25 Photon number 2 operators 0,20 operator corresponding to Distribution 0,15 the 2 possible phaseshift per atom in | g atom in | g & & 0,10 photon outcomes 0,05 0,00 0 1 2 3 4 5 6 7 8 Photon number, 11 11
Probe : weak measurement weak measurement Probe : Fixing the parameters of experiment Fixing the parameters of experiment • • Phaseshift Phaseshift per per photon : photon : • • Ramsey phase : : Ramsey phase Three well Three well distinct sets distinct sets 12 12
Probe : weak measurement weak measurement Probe : Fixing the parameters of experiment Fixing the parameters of experiment • • Phaseshift Phaseshift per per photon : photon : • • Ramsey phase: : Ramsey phase Quantum jumps well detected Quantum jumps well detected • • • • 13 13
Controler : real time : real time Controler estimation of field field state state estimation of Before weak measurement, , field described field described by by density matrix density matrix Before weak measurement • Weak measurement • Weak measurement Detected atom Detected atom : : outcome outcome « Ideal Ideal » situation: » situation: does does not not take into account take into account « imperfections the imperfections of of experimental set-up experimental set-up ! ! the 14 14
Controler : : field field state estimation state estimation Controler Difficulty : : atomic atomic source Difficulty source is is not not deterministic deterministic a ! ! 0,6 Poisson law law for for atom number per sample with average atom number per sample with average : n : n a 0,6 atom atom Poisson Most probable : no no atom atom in in sample sample Most probable : Two Two atoms atoms possible possible New New POVM POVM operators when operators when 2 2 atoms detected atoms detected 15 15
Controler : : field field state estimation state estimation Controler Difficulty : Difficulty : imperfect apparatus imperfect apparatus ! ! • 35 % • Detection efficiency Detection efficiency : : 35 % of of atoms atoms are are counted counted Unread measurements: : Unread measurements 16 16
Controler : : field field state estimation state estimation Controler Difficulty : Difficulty : imperfect apparatus imperfect apparatus ! ! • 35 % • Detection efficiency Detection efficiency : : 35 % of of atoms atoms are are counted counted Unread measurement Unread measurement proportion of atoms proportion of atoms in in |e |e & & detected in in |g |g & detected & • Limited interferometer Limited interferometer contrast contrast • Detection errors Detection errors 17 17
Controler : : field field state estimation state estimation Controler • Poisson • Poisson statistics statistics Difficulty : Difficulty : imperfect apparatus imperfect apparatus • Detection efficiency Detection efficiency • • • Detection errors Detection errors Assume 1 Assume 1 atom detected atom detected in state in state • • Was really the atom Was really the atom in in this this state? state? or ? or ? • • Was Was a second a second atom missed atom missed ? ? • If or • If so so, in , in which which state state was it was it ? ? or ? ? All conditional probabilities given by Bayes law, knowing calibrated imperfections I. Dotsenko et al. , Phys. Rev. A 80 , 013805 (2009) 18 18
Actuator : : field displacement field displacement Actuator Change photon number distribution distribution via via field displacement field displacement Change photon number Displacement operator : injection of coherent field in cavity amplitude of displacement : complex amplitude of microwave pulse I n experiment experiment : : I n α • α • real real only only • • phase phase is chosen is chosen to to be be 0 or 0 or $ $ , , with with respect to initial respect to initial field field (fixing (fixing sign sign of of displacement displacement) ) | α α | • Modulus via • Modulus | | is controled is controled via duration duration of of microwave microwave pulse pulse 19 19
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